Single Variable Calculus, Chapter 2, 2.2, Section 2.2, Problem 19

Evaluate the function $\displaystyle \lim \limits_{x \to 0} \frac{\sin x}{x + \tan x}$ at the given
numbers $x = \pm 1, \pm 0.5, \pm 0.2, \pm 0.1, \pm 0.05, \pm 0.01$ and guess the value of the limit, if it exists.

Substitute the given values of $x$


$\begin{equation} \begin{aligned} \begin{array}{|c|c|} \hline\\ x & f(x) \\ \hline\\ 1 & 0.329033 \\ 0.5 & 0.458209 \\ 0.2 & 0.493331 \\ 0.1 & 0.498333 \\ 0.05 & 0.499583 \\ 0.01 & 0.499983 \\ -0.01 & 0.499983 \\ -0.05 & 0.499583 \\ -0.1 & 0.498333 \\ -0.2 & 0.493331 \\ -0.5 & 0.458209 \\ -1 & 0.329033\\ \hline \end{array} \end{aligned} \end{equation}$


The table shows that as $x$ approaches 0 from left and right, the limit approaches a value of $\displaystyle \frac{1}{2}$
.


$\displaystyle \lim \limits_{x \to 0} \frac{\sin x}{x + \tan x} = \frac{\sin (0.000001)}{0.000001 + \tan (0.000001)} = \frac{1}{2}$